Question:

A sports team of $11$ students is to be constituted, choosing atleast $5$ from class and atleast $5$ from Class If there are $20$ students in each of these classes, in how many ways can the team be constituted?

Updated On: Jul 6, 2022
  • $20 \times \, ^{20}C_{5}$
  • $^{20}C_{5} \times\, ^{20}C_{6}$
  • $2\times\, ^{20}P_{5} \times\, ^{20}P_{6}$
  • $2\times\,^{20}C_{5} \times\, ^{20}C_{6}$
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The Correct Option is D

Solution and Explanation

No. of students in each class $= 20$ We have to select atleast $5$ students from each class. $\therefore$ No. of selection of sports team of $11$ students from each class is given in following table.
$\therefore$ Total number of ways of selecting a team of $11$ players $=(^{20}C_{5} \times\, ^{20}C_{6}) +\, (^{20}C_{6} \times\, ^{20}C_{5}) $ $= 2\times\, ^{20}C_{5} \times\,^{20}C_{6}$
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Concepts Used:

Permutations and Combinations

Permutation:

Permutation is the method or the act of arranging members of a set into an order or a sequence. 

  • In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point. 
  • A permutation is used in many events of daily life. It is used for a list of data where the data order matters.

Combination:

Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.

  • Combination refers to the combination of about n things taken k at a time without any repetition.
  • The combination is used for a group of data where the order of data does not matter.